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270CHAPTER 12. STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC
BARS IN TOR
x x
x
y
zT
L
free-body cut
Mt (x)
x
xMt (x) = T
free-body a
Mt (x) Mt (x + x)
free-body b torque diagram
T
x
T
x
Mt (x)
Mt (x) = T
Figure 12.5: Circular Bar Subjected to End Torque T
Applying conservation of angular momentum (i.e., summing moments
about the x-axis) in free-body a, we see that that the internal
torque Mt(x) is a constant for this case and equal to theapplied
external torque T. Hence any section between x and x + x will have
constant torqueapplied to it as shown in Figure 12.5 (free-body b).
Note that a positive torque on a right faceis in the +x direction
while the same positive torque is in the x direction on the left
face (torquemust be equal and opposite on opposite ends for moment
equilibrium to exist). This is the same typeof sign convention as
was used for positive tension in a truss member or for stresses.
The internaltorque diagram is also shown in Figure 12.5 and we see
that the internal torque, Mt(x), plots as aconstant value ofT. For
a more general torque loading as shown in Figure 12.4, the internal
torqueMt(x) would be a function ofx and the torque diagram would be
more complex; this case will bediscussed shortly.
Before developing a theory for how much a bar twists and its
resultant stress state under a
torque loading, it is very instructive to experimentally observe
the deformation pattern of a twistedbar as shown in the following
photograph. The bar has a solid circular cross-section and is
subjectedto equal and opposite torques as shown (provided by the
opposite twisting of each hand). Theinternal torque, Mt(x), must
therefore be constant throughout the bar. The undeformed bar
hasbeen marked with straight lines that run the length of bar as
well as circular lines around thecircumference as shown in the top
photograph.
Note that these lines form a pattern of squares on the surface
of the bar. After twisting (lowerphotograph), the straight lines
spiral around the bar and the circular lines remain circles. For
asmall area on the curved surface (say one of the squares), the
spiraling lines would be straight if thecurved surface were laid
out flat. The squares after twisting are now parallelograms, which
suggeststhat a shear stress has been applied to the squares.
To begin the development of a torsional theory, consider a
circular bar such as that shown inFigure 12.5 with the left end
fixed from rotation and a torque T applied at the right end.
Asdiscussed previously, the internal torque Mt = T will be a
constant through out the bar.
We scribe a line on the surface of the bar that runs the length
of the bar from 0 to a. After thetorque is applied, line 0a moves
to 0a. The displacement of point a can be described in
Cartesiancoordinates by the components uz and uy or in cylindrical
coordinates as u, as shown in Figure12.7. Looking at the end
cross-section where the torque is applied, we see that line b-a has
rotatedCCW to b-a by the angle . The angle represents the angle of
twist of the cross-section locatedat some point x relative to the
left end (x = 0) which is assumed fixed from rotation. Note: Do
